1 Polynomial ideals - MIT OpenCourseWare

1 MIT Algebraic techniques and semide nite optimization March 23, 2006. Lecture 13. Lecturer: Pablo A. Parrilo Scribe: ??? Today we introduce the rst basic elements of algebraic geometry, namely ideals and varieties over the complex numbers. This dual viewpoint ( ideals for the algebra, varieties for the geometry) is enormously powerful, and will help us later in the development of methods for solving Polynomial equations. We also present the notion of quotient rings, which are very natural when considering functions de ned on algebraic varieties ( , in Polynomial optimization problems with equality constraints). Finally, we begin our study of Groebner bases, by de ning the notion of term orders. A superb introduction to algebraic geometry, emphasizing the computational aspects, is the textbook of Cox, Little, and O'Shea [CLO97].

2 1 Polynomial ideals For notational simplicity, we use C[x] to denote the Polynomial ring in n variables C[x1 , .. , xn ]. Spe . cializing the general de nition of an ideal to a Polynomial ring, we have the following: De nition 1. A subset I C[x] is an ideal if it satis es: 1. 0 I. 2. If a, b I, then a + b I. 3. If a I and b C[x], then a b I. The two most important examples of Polynomial ideals for our purposes are the following: The set of polynomials that vanish in a given set S Cn , , I(S) := {f C[x] : f (a1 , .. , an ) = 0 (a1 , .. , an ) S}, is an ideal, called the vanishing ideal of S. The ideal generated by a nite set of polynomials {f1 , .. , fs }, de ned as f1 , .. , fs := {f | f = g1 f1 + + gs fs , gi C[x]}. (1). An ideal is nitely generated if it can be written as in (1) for some nite set of polynomials {f1.}

3 , fs }. An ideal is called principal if it can be generated by a single Polynomial . The intersection of two ideals is again an ideal. What about the union of ideals ? Example 2. In the univariate case ( , the Polynomial ring is C[x]), every ideal is principal. One of the most important facts about Polynomial ideals is Hilbert's niteness theorem: Theorem 3 (Hilbert Basis Theorem). Every Polynomial ideal in C[x] is nitely generated. We will present a proof of this after learning about Groebner bases. From the computational viewpoint, two very natural questions about ideals are the following: Given a Polynomial p(x), how to decide if it belongs to a given ideal? How to nd a convenient representation of an ideal? What does convenient mean? 13 1. 2. y 0. 2. -2. 2. 1. 1. x -2 -1 1 2 0. -1. -1. -2. -2. -2 0. 2.

4 Figure 1: Two algebraic varieties. The one on the left is de ned by the equation (x2 + y 2 1)(3x +. 6y 4) = 0. The one on the right is a quartic surface, de ned by 1 x2 y 2 2z 2 + z 4 = 0. 2 Algebraic varieties An (a ne) algebraic variety is the zero set of a nite collection of polynomials (see formal de nition below). The word a ne here means that we are working in the standard a ne space, as opposed to projective space, where we identify x, y Cn if x = y for some = 0. De nition 4. Let f1 , .. , fs be polynomials in C[x]. Let the set V be V(f1 , .. , fs ) := {(a1 , .. , an ) Cn : fi (a1 , .. , an ) = 0 1 i s}. We call V(f1 , .. , fs ) the a ne variety de ned by f1 , .. , fs . A simple example of a variety is a (complex) a ne subspace, that corresponds to the vanishing of a nite collection of a ne polynomials.

5 A few additional examples of varieties are shown in Figure 1. It is not too hard to show that nite unions and intersections of algebraic varieties are again algebraic varieties. What about the in nite case? Remark 5. Recall our previous encounter with the Zariski topology, whose closed sets where de ned to be the algebraic varieties, , the vanishing set of a nite set of Polynomial equations. To prove that this is actually a topology, we need to show that arbitrary intersections of closed sets are closed. Hilbert's basis theorem precisely guarantees this fact. Perhaps the most natural question about algebraic varieties is the following: Given a variety V , how to decide it is nonempty? Let's start connecting ideals and varieties. Consider a nite set of polynomials {f1 , .. , fs }. We already know how to generate an ideal, namely f1.

6 , fs . However, we can also look at the corre . sponding variety V(f1 , .. , fs ). Since this variety is a subset of Cn , we can form the corresponding vanishing ideal, I(V(f1 , .. , fs )). How do these two ideals related to each other? Is it always the case that f1 , .. , fs = I(V(f1 , .. , fs )), and if it is not, what are the reasons? The answer to these questions (and more) will be given by another famous result by Hilbert, known as the Nullstellensatz. 13 2. 3 Quotient rings Whenever we have an ideal in a ring, we can immediately de ne a notion of equivalence classes, where we identify two elements in the ring if and only if their di erence is in the ideal. Example 6. Recall that a simple example of an ideal in the ring Z was the set of even integers. By identifying two integers if their di erence is even, we partition Z into two equivalence classes, namely the even and the odd numbers.

7 More generally, if the ideal is given by the integer multiples of a given number m, then Z can be partitioned into m equivalence classes. We can do this for the Polynomial ring C[x], and any ideal I. De nition 7. Let I C[x] be an ideal, and let f, g C[x]. We say f and g are congruent modulo I, written f g mod I, if f g I. It is easy to show that this is an equivalence relation, , it is re exive, symmetric, and transi . tive. Thus, this partitions C[x] into equivalence classes, where two polynomials are the same if their di erence belongs to the ideal. This allows us to de ne the quotient ring: De nition 8. The quotient C[x]/I is the set of equivalence classes for congruence modulo I. The quotient C[x]/I inherits the ring structure of C[x], with the natural operations. Thus, with these operations now de ned between equivalence classes, C[x]/I becomes a ring, known as the quotient ring.

8 Quotient rings are particularly useful when considering a Polynomial function p(x) over the algebraic variety de ned by gi (x) = 0. Notice that if we de ne the ideal I = gi , then any Polynomial q that is congruent with p modulo I takes exactly the same values in the variety. 4 Monomial orderings In order to begin studying nice bases for ideals , we need a way of ordering monomials. In the univariate case, this is straightforward, since we can de ne xa xb as being true if and only if a > b. In the multivariate case, there are a lot more options. We also want the ordering structure to be consistent with Polynomial multiplication. This is formal . ized in the following de nition. De nition 9. A monomial ordering on C[x] is a relation on Zn+ ( , the monomial exponents), such that: 1. The relation is a total ordering.

9 2. If , and Zn+ , then + + . 3. The relation is a well ordering (every nonempty subset has a smallest element). One of the simplest examples of a monomial ordering is the lexicographic ordering, where lex if the left most nonzero entry of is positive. We will see some other examples of monomial orderings later in the course. References [CLO97] D. A. Cox, J. B. Little, and D. O'Shea. ideals , varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer, 1997. 13 3.